Finite Group Databasebeta

← #11896067

Subgroups — #11896067

Order 1152.

782 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596055C211yesmaximal, characteristic
2C3×Q8×Dic6C211yesmaximal, characteristic, complemented
3C3×#5024C211yesmaximal, characteristic, complemented
4#10590385C211yesmaximal, characteristic, complemented
5#10590383C212yesmaximal, complemented
6#10590368C212yesmaximal
7C4×#3717C211yesmaximal, characteristic
8#10590307C212yesmaximal, complemented
9#10590300C211yesmaximal, characteristic
10#10590294C212yesmaximal
11#10590288C211yesmaximal, characteristic, complemented
12#83704—31nomaximal
13#79981—31nomaximal
14C32×C4×Q8C2211yescharacteristic
15#64155—21no—
16#64107—22no—
17#64101—21no—
18#64084C2212yes—
19C4×C4⋊(C3⋊D3)C2211yescharacteristic
20#64078C2211yescharacteristic
21C4×#634C2212yes—
22C3×Q8×Dic3—21no—
23C3×#888—21no—
24C3×#888—22no—
25C3×#869C2211yescharacteristic
26C3×#869C2212yes—
27C3×C4×Dic6C2211yescharacteristic
28C3×C4×Dic6C2212yes—
29C2×#3719C2212yes—
30C2×#3717C2211yescharacteristic
31C3×C4⋊(C3⋊C8)C2211yescharacteristic
32C3×C4⋊(C3⋊C8)C2212yes—
33C3×#803C2212yes—
34C3×C4×C3⋊C8C2211yescharacteristic
35#63573C2212yes—
36#63573C2211yescharacteristic
37#63567C2212yes—
38#63567C2212yes—
39#63567C2212yes—
40#63566C2211yescharacteristic
41#63564C2211yescharacteristic
42#63564C2212yes—
43C3×Q82—31no—
44#5584—31no—
45#5584—31no—
46#5584—61no—
47Q8×Dic6D311yescharacteristic, complemented
48C3×#446—31no—
49#5094—32no—
50#5093—31no—
51#5089—32no—
52#5084—31no—
53#5031—31no—
54#5030—32no—
55#5028—32no—
56C4×#215—31no—
57#5024D311yescharacteristic, complemented
58#4851—32no—
59#4849—31no—
60#4845—32no—
61#4842—31no—
62#4716—32no—
63#4714—32no—
64C4×#204—31no—
65#4681—31no—
66C2×C32×Q8D411yescharacteristic
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×#634C2312yes—
70C2×C3×Dic6—22no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2311yescharacteristic
73C32×C4⋊C4D411yescharacteristic
74C32×C4⋊C4D412yes—
75C32×C42C2312yes—
76C32×C42C2311yescharacteristic
77#3752—21no—
78#3752—22no—
79#3752—22no—
80C4⋊(C3⋊Dic3)—21no—
81C4⋊(C3⋊Dic3)—21no—
82C4⋊(C3⋊Dic3)C2311yescharacteristic
83#3750—22no—
84C4×C3⋊Dic3—22no—
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3C2311yescharacteristic
88C3×C4⋊Dic3C2312yes—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—21no—
91C3×C4×Dic3—22no—
92C3×C4×Dic3—21no—
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2312yes—
95C2×C3×C3⋊C8C2311yescharacteristic
96#3719—24no—
97#3717—22no—
98#3369—91noSylow
99C3×C4⋊Q8—31no—
100C3×C4⋊Q8—32no—
101C3×C4⋊Q8—61no—
102C3×C4⋊Q8—62no—
103C3×C4×Q8—21no—
104C3×C4×Q8—31no—
105C3×C4×Q8—32no—
106C3×C4×Q8—61no—
107C3×C4×Q8C2×D311yescharacteristic
108C3×C4×Q8C2×D311yescharacteristic
109#946—62no—
110#946—61no—
111#946—61no—
112Q8×Dic3—21no—
113C2×#216—32no—
114C2×#215—31no—
115C2×Dic12—32no—
116C2×#204—31no—
117#898—64no—
118#898—62no—
119#898—62no—
120#892—62no—
121#892—61no—
122#892—61no—
123#888—21no—
124#888—22no—
125#875—32no—
126#875—32no—
127#875—62no—
128C4×D12—31no—
129C4×D12—31no—
130C4×D12—61no—
131#869—31no—
132#869—31no—
133#869—61no—
134#869C2×D311yescharacteristic
135#869C2×D312yes—
136C4×Dic6—32no—
137C4×Dic6—32no—
138C4×Dic6—62no—
139C4×Dic6C2×D311yescharacteristic
140C4×Dic6C2×D312yes—
141C3×C4⋊C8—31no—
142C3×C4⋊C8—32no—
143C3×#97—32no—
144C3×C4×C8—31no—
145#835—31no—
146#821—31no—
147#821—32no—
148#817—31no—
149#817—32no—
150#816—31no—
151#816—32no—
152#816—32no—
153#816—32no—
154#810—32no—
155#810—32no—
156#810—32no—
157#809—31no—
158#809—32no—
159#808—31no—
160#808—32no—
161C4⋊(C3⋊C8)C2×D312yes—
162C4⋊(C3⋊C8)C2×D311yescharacteristic
163#803C2×D312yes—
164C4×C3⋊C8C2×D311yescharacteristic
165C22×C3⋊D3—21no—
166C32×Q8—22no—
167C2×C32×C4C2×D412yes—
168C2×C32×C4C2×D412yes—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2411yescharacteristic
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—22no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176C4×C3⋊D3—42no—
177#634—22no—
178#634—22no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—21no—
182C3×Dic6—21no—
183C3×Dic6—82no—
184C3×Dic6—81no—
185C3×Dic6#4112yes—
186C3×C3⋊C8—21no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—21no—
189Q82—31no—
190#556—91no—
191#489—92no—
192#487—91no—
193#476—91no—
194#475—92no—
195#464—92no—
196#462—91no—
197#446—31no—
198#441—92no—
199C4×SD16—91no—
200C2×C3×Q8—21no—
201C2×C3×Q8—31no—
202C2×C3×Q8—62no—
203C2×C3×Q8—61no—
204C2×C3×Q8D1211yescharacteristic
205C2×C3×Q8#6711yescharacteristic
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—61no—
209C2×C4×D3—62no—
210C2×C4×D3—61no—
211C2×C4×D3—61no—
212C2×Dic6—22no—
213C2×Dic6—21no—
214C2×Dic6—32no—
215C2×Dic6—32no—
216C2×Dic6—62no—
217C2×Dic6C22×D311yescharacteristic
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—32no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—62no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—62no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4D1211yescharacteristic
230C3×C4⋊C4D1212yes—
231C3×C4⋊C4#6711yescharacteristic
232C3×C4⋊C4#6712yes—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42—62no—
236C3×C42—61no—
237C3×C42C22×D311yescharacteristic
238C3×C42C22×D312yes—
239C3×C42C22×D311yescharacteristic
240C3×C42C22×D312yes—
241#216—64no—
242#215—62no—
243(C2×D3)⋊C4—62no—
244(C2×D3)⋊C4—64no—
245(C2×D3)⋊C4—64no—
246(C2×D3)⋊C4—61no—
247(C2×D3)⋊C4—62no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—61no—
250(C2×D3)⋊C4—62no—
251(C2×D3)⋊C4—62no—
252C4⋊Dic3—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—61no—
257C4⋊Dic3—62no—
258C4⋊Dic3—62no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3—61no—
262C4⋊Dic3—61no—
263C4⋊Dic3C22×D312yes—
264C4⋊Dic3C22×D311yescharacteristic
265Dic3⋊C4—22no—
266Dic3⋊C4—21no—
267Dic3⋊C4—64no—
268Dic3⋊C4—62no—
269Dic3⋊C4—62no—
270C4×Dic3—22no—
271C4×Dic3—21no—
272C4×Dic3—64no—
273C4×Dic3—62no—
274C4×Dic3—62no—
275C2×C3⋊C8C22×D311yescharacteristic
276C2×C3⋊C8C22×D311yescharacteristic
277C2×C3⋊C8C22×D312yes—
278Dic12—64no—
279#204—62no—
280C22×C32C22×D411yescharacteristic
281C2×C3⋊D3—41no—
282C2×C3⋊D3—41no—
283C2×C3⋊D3—41no—
284C32×C4—22no—
285C32×C4—22no—
286C32×C4—21no—
287C32×C4#13712yes—
288C32×C4C2×#4111yescharacteristic
289C32×C42−1+411yescharacteristic
290C3⋊Dic3—42no—
291C3⋊Dic3—41no—
292C3×Dic3—22no—
293C3×Dic3—41no—
294C3×Dic3—42no—
295C2×Q16—92no—
296C2×SD16—91no—
297C4⋊Q8—32no—
298C4⋊Q8—31no—
299C4⋊Q8—61no—
300C4⋊Q8—62no—
301C4⋊Q8—91no—
302#126—182no—
303#124—92no—
304C22⋊Q8—181no—
305C4×Q8—31no—
306C4×Q8—32no—
307C4×Q8—61no—
308C4×Q8—92no—
309C4×Q8D3211yescharacteristic
310C4×D4—91no—
311#117—181no—
312#106—91no—
313#106—92no—
314C4⋊C8—31no—
315C4⋊C8—32no—
316#103—92no—
317#103—92no—
318#103—92no—
319#103—91no—
320#102—91no—
321#102—92no—
322#97—32no—
323C4×C8—31no—
324C22×D3—62no—
325C22×D3—61no—
326C22×D3—61no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—32no—
330C3×Q8—42no—
331C3×Q8—61no—
332C3×Q8—242no—
333C3×Q8—241no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—61no—
340C2×C3×C4—62no—
341C2×C3×C4—61no—
342C2×C3×C4C2×D1212yes—
343C2×C3×C4C2×D1212yes—
344C2×C3×C4C2×D1211yescharacteristic
345C2×C3×C4C2×D1211yescharacteristic
346C2×C3×C4C2×#6712yes—
347C2×C3×C4C2×#6712yes—
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C23×D311yescharacteristic
351C2×C3×C4C23×D311yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—21no—
355C2×Dic3—62no—
356C2×Dic3—64no—
357C2×Dic3—61no—
358C2×Dic3—62no—
359C2×Dic3—61no—
360C2×Dic3—62no—
361D12—61no—
362D12—61no—
363D12—61no—
364D12—61no—
365D12—121no—
366D12—121no—
367C4×D3—124no—
368C4×D3—122no—
369C4×D3—122no—
370Dic6—21no—
371Dic6—62no—
372Dic6—62no—
373Dic6—62no—
374Dic6—62no—
375Dic6—82no—
376Dic6—81no—
377Dic6—122no—
378Dic6—122no—
379Dic6#23912yes—
380C3×C8—61no—
381C3×C8—62no—
382C3×C8—61no—
383C3⋊C8—21no—
384C3⋊C8—21no—
385C3⋊C8—22no—
386C2×C32#54811yescharacteristic
387C2×C32C2×#13711yescharacteristic
388C2×C32#57711yescharacteristic
389C3⋊D3—81no—
390C2×Q8—31no—
391C2×Q8—62no—
392C2×Q8—61no—
393C2×Q8—92no—
394C2×Q8#62611yescharacteristic
395C2×D4—91no—
396C22×C4—181no—
397Q16—184no—
398SD16—182no—
399C2×C8—31no—
400C2×C8—31no—
401C2×C8—32no—
402C4⋊C4—31no—
403C4⋊C4—32no—
404C4⋊C4—62no—
405C4⋊C4—61no—
406C4⋊C4—62no—
407C4⋊C4—61no—
408C4⋊C4—91no—
409C4⋊C4—181no—
410C4⋊C4—182no—
411C4⋊C4—181no—
412C4⋊C4#62611yescharacteristic
413C4⋊C4#62612yes—
414C22⋊C4—182no—
415C22⋊C4—181no—
416C22⋊C4—182no—
417C42—62no—
418C42—61no—
419C42—182no—
420C42C2×D3211yescharacteristic
421C42C2×D3212yes—
422C22×C3—21no—
423C22×C3C22×D1211yescharacteristic
424C22×C3C22×#6711yescharacteristic
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—122no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—41no—
446C3×C4—62no—
447C3×C4—122no—
448C3×C4—121no—
449C3×C4#90912yes—
450C3×C4#94212yes—
451C3×C4C2×#23911yescharacteristic
452C3×C4#100711yescharacteristic
453C3×C4#100711yescharacteristic
454C3×C4D3×#4111yescharacteristic
455Dic3—22no—
456Dic3—41no—
457Dic3—42no—
458Dic3—124no—
459Dic3—122no—
460Dic3—121no—
461Dic3—121no—
462Dic3—122no—
463Dic3—122no—
464C32#336911yescharacteristic, complemented, Sylow
465C23—181no—
466Q8—22no—
467Q8—32no—
468Q8—61no—
469Q8—182no—
470Q8—182no—
471Q8—242no—
472Q8—241no—
473D4—181no—
474D4—181no—
475C2×C4—61no—
476C2×C4—62no—
477C2×C4—61no—
478C2×C4—181no—
479C2×C4—182no—
480C2×C4—362no—
481C2×C4C2×#62612yes—
482C2×C4C2×#62612yes—
483C2×C4C2×#62611yescharacteristic
484C2×C4C2×#62611yescharacteristic
485C2×C4C22×D3211yescharacteristic
486C8—61no—
487C8—61no—
488C8—62no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3#557611yescharacteristic
493C2×C3C2×#90911yescharacteristic
494C2×C3#575111yescharacteristic
495C2×C3C2×#94211yescharacteristic
496C2×C3#581711yescharacteristic
497C2×C3#583611yescharacteristic
498D3—242no—
499D3—241no—
500D3—241no—
501C22—361no—
502C22—361no—
503C22—361no—
504C22C22×#62611yescharacteristic
505C4—22no—
506C4—21no—
507C4—22no—
508C4—62no—
509C4—122no—
510C4—121no—
511C4—362no—
512C4—361no—
513C4#6383312yes—
514C4#6431611yescharacteristic
515C4D3×#23911yescharacteristic
516C3—21no—
517C3#7998111yescharacteristic, complemented
518C3#8370411yescharacteristic, complemented
519C2—721no—
520C2C2×#6383311yescharacteristic
521C2#1059534211yescharacteristic
522C2#1059544711yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenterC4×Q8D32
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×Q8×Dic6C2semidirect
3. C3×#5024C2semidirect
4. #10590385C2semidirect
5. #10590383C2semidirect
8. #10590307C2semidirect
11. #10590288C2semidirect
47. Q8×Dic6D3semidirect
57. #5024D3semidirect
464. C32#3369semidirect
517. C3#79981semidirect
518. C3#83704semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#105960552
C3×Q8×Dic62
C3×#50242
#105903852
#105903832
#105903682
C4×#37172
#105903072
#105903002
#105902942
#105902882
#837043
#799813